In the table below we give for each conjugacy class of \(M_{24}\)  its cycle length, the associated genus zero subgroup, and when possible the Niemeier root system of which the corresponding genus zero group is conjugated to that of the conjugacy class, see [1]} and [2]}.
\(\begin{array}{|c|l|l|l|| c | l | l | l |}\hline 1A & 1^{24} & 2- & A_1^{24} & 7AB & 1^37^3 & 14+7 & D_8^3 \\2A & 1^82^8 & 4- & A_3^8 & 8A & 1^2.2.4.8^2 & 16- & A_{15}D_9 \\2B & 2^{12} & 4|2- & A_1^{24} & 10A & 2^210^2 & 20|2+5 & D_6^4 \\3A & 1^63^6 & 6+3 & D_4^6 & 11A & 1^211^2 & 22+11 & D_{12}^2 \\3B & 3^8 & 6|3 & A_1^{24} & 12A & 2.4.6.12 & 24|2 + 3 & . \\4A & 2^44^4 & 8|2- & A_3^8 & 12B & 12^2 & 24|12- & A_1^{24} \\4B & 1^42^24^4 & 8- & A_2^7D_5^2 & 14AB & 1.2.7.14 & 28+7 & . \\4C & 4^6 & 8|4- & A_1^{24} & 15AB & 1.3.5.15 & 30+3,5,15 & . \\5A & 1^45^4 & 10+5 & D_6^4 & 21AB & 3.21 & 42|3+7 & D_8^3 \\6A & 1^22^23^26^2 & 12+3 & . & 23AB & 1.23 & 46+23 & D_{24} \\6B & 6^4 & 12|6- & A_1^{24} & & & & \\\hline \end{array}\) 
